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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

20 September 2026

9 essays on the shape is the unknown, the curve as an equation, motion that stops, more than one input, several legs, one platform, members that pull and as built

The shape is the unknown

A twist steadies what it cannot tighten

A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.

7 figures

A rectangular hyperbola, compiled from a multiple of its equation. The machine compiled from p · (1 + x² + y²) for a rectangular hyperbola, with the translators — the parallelograms that carry a direction from where it is produced to where it is needed — in their own colour. The factor 1 + x² + y² is at least one at every real point, so every equation here vanishes on exactly the same curve. The machines do not agree: 20 bars at p, 75 bars at p · (1 + x² + y²), 144 bars at p · (1 + x² + y²)². This one solves 29 positions over an arc of 0.508 radians, and at every one of them the original polynomial reads 4.93e-14.

The curve as an equation

The price is on the equation

A line costs five bars. The same line, written as its own equation multiplied by a factor that is never zero, costs fifty — and the machine compiled from the longer equation draws the same line just as exactly. Every cost this field quotes belongs to a polynomial and not to a curve, and the cheapest equation of a given curve is a quantity nobody here has.

7 figures

Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both.

The curve as an equation

Two circles for the price of one

Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.

5 figures

Eight contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 8 pins are in contact with the disc at this instant, and each one's line is the common normal, which passes through the pitch point on the pin circle. A pin can only push, so the sense in which it turns the disc is decided by which side of the disc's own centre its normal passes: 4 of the contacts turn it one way and 4 the other, with the largest arm in each sense 42.2 and 45.0 on a pitch offset of 55.0. A pair with one contact has no such choice, which is the whole of why two identical rotors cannot drive each other.

The shape is the unknown

What a second contact is for

Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has eight to eleven contacts at once instead of one, and never loses more than thirty per cent of the arm its geometry allows.

6 figures

Motion that stops

The disc decides the pin count

A Geneva's pin count is usually bounded by the slots: p indexes must not overlap, so fewer than 2n/(n−2) pins fit. The other half of the mechanism has its own inequality and nobody had measured it. The locking disc must be cut away wherever the wheel passes through it, that cut-away is wider than the index sweep at every slot count, and it is the binding condition everywhere — one pin only, from four slots upward.

5 figures

What each branch carries, against what the engine delivers. The power in the variator's branch and in the straight path, as multiples of the engine's own, for a planetary of K = 1.40. Both are ratios of powers, so the load cancels and nothing here is a force: the split comes from requiring the gearset to be lossless at every admissible set of speeds, which fixes the torques at 1 : K : −(1+K). The variator carries v/(K − v) and the straight path K/(K − v), and both run away at the pole. The variator first carries the engine's whole power at v = 0.700, which is exactly half the way to the pole — and the straight path is already carrying more than the engine everywhere past nought, flowing the other way through the planetary. That excess is the circulation.

More than one input

The power that goes round twice

Sliding a power split's travel towards its pole buys ratio span for nothing, on the kinematics. It is not for nothing. The variator's own branch carries v/(K−v) of the engine's power and overtakes it at exactly half the way to the pole, and the tolerance trim the span was computed from is not reached until the variator is rated for six times the engine — which no machine is.

5 figures

Several legs, one platform

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

4 figures

The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off.

Members that pull

Which walls a strand is held by

A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

6 figures

The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer.

As built

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

5 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

17 September 2026

17 essays on the problem backwards, what a joint is, members that pull, more than one input, many of one thing, several legs, one platform, contacts that only push, out of the plane, motion that stops, how many answers, wheels, and where they may not go, the chain before the lengths, the motion, not the mechanism, prescribed motion, linkages, the paths points trace and what can move

Where a pin becomes a slide — the problem backwards
A coupling that only translates — what a joint is
A strand in a tube — members that pull
Sliding the travel across the pole — more than one input
The count says how many and not where — many of one thing
The dead yaw is a design choice — several legs, one platform
The hold is in the corners — contacts that only push
The lines a leg turns about and the lines it is pushed along — out of the plane
When the index law becomes a choice — motion that stops
Where a slide puts the rest of the degree — how many answers
A bearing is a planetary with no teeth — wheels, and where they may not go
A slide turns nothing — the chain before the lengths
The frame seen from the coupler — the motion, not the mechanism
The time a crossover takes — prescribed motion
A quick return that cuts evenly — linkages
The dip that buys the dwell — the paths points trace
Every change point lies flat — what can move

15 September 2026

12 essays on teeth, links with a width, the motion, not the mechanism, members that pull, out of the plane and many of one thing

The angle the standard left free — teeth
The gap is a straight line in the metal — links with a width
The linkage, put back from two curves — the motion, not the mechanism
The wrap that walks along the axis — members that pull
What a leg of three joints leaves free — out of the plane
Which diagonal rigidifies a grid — many of one thing
A tooth that lives on a sphere — teeth
The regions overlap and the parts never meet — links with a width
The two numbers are the curves' own — the motion, not the mechanism
A link may be bent — links with a width
The mesh with one curvature reversed — teeth
Contact that runs along the tooth — teeth

14 September 2026

11 essays on motion that stops, more than one input, prescribed motion, several legs, one platform, the shape is the unknown, the paths points trace, how many answers, linkages and what can move

Two pins and no dwell at all — motion that stops
A bounded ratio made unbounded — more than one input
A flat face on an arm is worse — prescribed motion
A ratchet with no teeth — motion that stops
One placement of every placement — several legs, one platform
The clearance that is the seal — the shape is the unknown
The kind is decided before the lengths are — the paths points trace
The mesh inside keeps the half — how many answers
Three rotations, and four benches — linkages
A length error is undone by its own size — what can move
The right angle as a tolerance — what can move

12 September 2026

13 essays on the paths points trace, prescribed motion, linkages, how many answers, what can move, the shape is the unknown and several legs, one platform

Where three machines keep one area — the paths points trace
An arm is an offset that grows with the lift — prescribed motion
Four kinds of slider-crank — linkages
Nine times through each circular point — how many answers
The flattest dwell is not the longest — the paths points trace
A drag link ahead of a crank-rocker — linkages
A null space of fifteen is not noise — the paths points trace
A parallelogram a micron wrong — what can move
A roller in a groove changes walls with the speed — prescribed motion
Every rational gear ratio has a degree — how many answers
Rotors that mesh and cannot drive each other — the shape is the unknown
Two orientations no position can rescue — several legs, one platform
Nine bars that ought to be rigid — what can move

11 September 2026

18 essays on the paths points trace, prescribed motion, how many answers, several legs, one platform, linkages and what can move

The curve the other assembly draws — the paths points trace
A point the machine never reaches — the paths points trace
A sextic that comes apart — the paths points trace
A symmetric curve from a lopsided machine — the paths points trace
An offset trades the rise for the return — prescribed motion
Never three circuits — how many answers
Round a cusp into another assembly — several legs, one platform
A degree counted on a line — how many answers
A flat face asks for a convex cam — prescribed motion
Eight kinds of four-bar — linkages
The area a coupler point encloses — the paths points trace
The smallest parallel robot — several legs, one platform
A cam that holds its follower both ways — prescribed motion
A swing and a time ratio — linkages
The curve nobody eliminates — how many answers
What a fourth leg buys — several legs, one platform
A yaw that is singular everywhere — several legs, one platform
One freedom, and a motion that never repeats — what can move

5 September 2026

50 essays on numbers that were measured, prescribed motion, several legs, one platform, linkages, out of the plane, teeth, as built, the motion, not the mechanism, more than one input, contacts that only push, wheels, and where they may not go, the problem backwards, links with a width, members that pull, the curve as an equation, the chain before the lengths, one path to the tool and drawn wrongly

A dimension is a measurement — numbers that were measured
The matrix a calibration inverts — numbers that were measured
The direction no protractor can see — numbers that were measured
The coordinates the site already had — numbers that were measured
A ruler and a protractor — numbers that were measured
How many poses are enough — numbers that were measured
Where a calibration should measure — numbers that were measured
A lift is a size and a law is a shape — prescribed motion
A platform that measures itself — several legs, one platform
What another measurement is worth — numbers that were measured
Four indices, four answers — numbers that were measured
Grashof is a shape test — linkages
Bennett's condition is a ratio — out of the plane
The module is a size, the ratio is a shape — teeth
The pose the machine cannot reach — numbers that were measured
A band with a direction in it — as built
A curvature is a size with a minus sign — the motion, not the mechanism
Every length wrong, every reading right — numbers that were measured
The transmission angle has no size — more than one input
A cone has no size — contacts that only push
A wheel that cannot report its radius — wheels, and where they may not go
The instrument's error, multiplied — numbers that were measured
What a synthesis assumes it knows — the problem backwards
Where the boundary moved again — as built
A body is all size — links with a width
A drum is a size, a wrap is a shape — members that pull
Reading a residual — numbers that were measured
A compiled machine and its own scale — the curve as an equation
A parameter the model has not got — numbers that were measured
A graph has no numbers at all — the chain before the lengths
One set of lengths, two machines — numbers that were measured
Three machines, one curve — numbers that were measured
An arm's parameters and its poses — one path to the tool
Nine parameters, two of them invisible — numbers that were measured
Six things a measurement cannot tell you — drawn wrongly
A machine that measures itself — numbers that were measured
Where a length comes from — numbers that were measured
Where the two analyses cross — numbers that were measured
The same part, dimensioned twice — numbers that were measured
Which feature to hold tight — numbers that were measured
The common normal, and where it is — numbers that were measured
A number that runs away — numbers that were measured
The chart breaks, the machine does not — numbers that were measured
Six per joint is two too many — numbers that were measured
Which numbers have a size — numbers that were measured
A count is neither — numbers that were measured
A calibration is a synthesis with more equations — numbers that were measured
Two instruments disagree about the worst — numbers that were measured
What a model is allowed to change — numbers that were measured
What this field cannot measure — numbers that were measured

1 September 2026

20 essays on links with a width, prescribed motion, the problem backwards, many of one thing, what a joint is, drawn wrongly, one path to the tool and as built

A link that takes up room — links with a width
A gap is a number — links with a width
A shape with a dent in it — links with a width
A gap with corners in it — links with a width
A sweep that missed nothing — links with a width
Two bars that have to cross — links with a width
A follower needs a face — prescribed motion
A plane is a colour — links with a width
The room a machine sweeps — links with a width
The hole the machine needs — links with a width
The crank that cannot turn all the way — links with a width
Free space comes in pieces — links with a width
A pin is not a point — links with a width
A defect that is not kinematic — the problem backwards
A stack that has to fit — many of one thing
The block in the guide has a length — what a joint is
Six things a body is not — drawn wrongly
The arm that hits itself — one path to the tool
Where the boundary moved — as built
A clearance inside a tolerance box — as built

31 August 2026

20 essays on the curve as an equation, the paths points trace, how many answers, what can move, the problem backwards, the chain before the lengths, drawn wrongly and as built

A demand that is an equation — the curve as an equation
Every curve is a sum of cosines — the curve as an equation
A circle costs one term — the curve as an equation
Four bars that add two angles — the curve as an equation
A parallelogram carries an angle, and only so far — the curve as an equation
Three linkages, one equation — the paths points trace
Doubling is cheaper than adding — the curve as an equation
The machine, compiled — the curve as an equation
Five bars for a line, four hundred for a quintic — the curve as an equation
The equation a four-bar satisfies — how many answers
The proof drew more than the curve — the curve as an equation
A bar between two midpoints — the curve as an equation
One freedom and four hundred links — what can move
Where the machine stops being the function — the curve as an equation
Exact costs more than close — the curve as an equation
Prescribing a curve rather than points — the problem backwards
What universality is worth — the curve as an equation
A machine with one dyad in it — the chain before the lengths
Six things a compiled linkage is not — drawn wrongly
Exactness a micron destroys — as built

30 August 2026

20 essays on what a joint is, several legs, one platform, what can move, out of the plane, wheels, and where they may not go, drawn wrongly, one path to the tool and as built

A joint is a surface that slides on itself — what a joint is
Six, and no others — what a joint is
The count cannot tell a pin from a slide — what a joint is
Twelve kinds of freedom — what a joint is
Almost nothing is a group — what a joint is
What a point sees — what a joint is
A chain multiplies — what a joint is
Three legs and one plane — several legs, one platform
Four joints that give a group, and four that do not — what a joint is
Legs intersect — what a joint is
The freedom that is a set — what can move
Two planes meeting in a line — what a joint is
Compose two positions and see where you land — what a joint is
A name for each overconstraint — out of the plane
The instrument that is not a derivative — what a joint is
A higher pair has no group — what a joint is
One bracket, two subjects — wheels, and where they may not go
Six things a joint is not — drawn wrongly
The arm that is a group — one path to the tool
The pair a catalogue sells — as built

28 August 2026

20 essays on the chain before the lengths, how many answers, what can move, the problem backwards, out of the plane, drawn wrongly, one path to the tool and as built